Applied Mathematical Sciences Vol. 8, 2014, no. 134, 6725 - 6732 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.49687 Axiomatic Set Theory and Unincreasable Infinity Raffaella Cutolo, Ulderico Dardano and Virginia Vaccaro Dipartimento di Matematica e Applicazioni \R. Caccioppoli" Universit`adegli Studi di Napoli \Federico II" Via Cintia, Complesso Monte S. Angelo, 80126 Naples, Italy Copyright c 2014 Raffaella Cutolo, Ulderico Dardano and Virginia Vaccaro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we propose a reflection on the use of axiomatic set theory as a fundamental tool to address the foundational issues of mathemat- ics. In particular, we focus on the key concept of infinity, indeed the strongest kind of infinity which may be conceived (what is referred to as \the Absolute" by Cantor), as we aim to show how the point of view of- fered by a specific set-theoretical framework allows us to deal with such a paradoxical notion in a completely safe manner. For this purpose, we shall introduce NBG set theory and discuss its consistency. We assume the reader is familiar with ZFC set theory (see for example [2] or [3]). Mathematics Subject Classification: 03E99 Keywords: Set theory, Absolute infinite \I have no doubt that, as we pursue this path [of generating num- bers] even further, we shall never reach a boundary that cannot be crossed; but that we shall also never achieve even an approximate conception of the Absolute." [G. Cantor] 1 Introduction Set theory may be regarded as \the mathematical science of the infinite”. Since ancient times, infinity has been a key part of mathematics, though its use has 6726 Raffaella Cutolo, Ulderico Dardano and Virginia Vaccaro often been considered harmful and it was thought of in the form of potential infinite, i.e., an unending process. Many researchers, including for example Dedekind, reflected on infinity as an indispensable tool of science and engi- neering; but only with the revolution due to the work of Georg Cantor (1845 - 1918) infinity became actual infinite, object of study and mathematical matter, instead of a purely metaphysical concept. Through a really elementary and ingenious idea (the so-called diagonal ar- gument), in 1874 Cantor realized that not all infinities are equal; in particular, the set of real numbers is not \countable", that is, equinumerous to the set of natural numbers. Counting the real numbers allowed him to discover that there exist many numbers which are not root of any polynomial of positive degree over the integers. This gave rise to \naive" set theory. Cantor pointed out the significance of one-to-one functions and introduced the notion of car- dinality of a set. Cantor's discovery of \uncountable" sets was one of the most unexpected events in the history of mathematics, and the subsequent formu- lation of his theory of transfinite numbers led to consider him as the founder of the mathematical theory of the infinite. He found an increasing chain of infinities marking the levels of the transfinite and delimited by what he termed \the Absolute", which he associated mathematically with the whole sequence of these infinities and metaphysically with God. But he suppressed all that pertains to the order of his personal motivations, he realized that the diagonal argument applies to any set and in 1891 stated his fundamental general result establishing that any set has \more" subsets than members. It follows, in turn, that there is no largest set. Therefore, there is no such thing as the \set of all sets"; in fact, the set of all sets, if it existed, would necessarily be the largest set. Furthermore, in 1901 Russell discovered his paradox: the set of all sets x such that x2 = x cannot exist. This reinforced the idea that mathematics needs more secure foundations, leading to the development of axiomatic set theories. The first axiomatization of set theory was published by Zermelo in 1908. Zermelo shifted the focus away from the transfinite numbers to an abstract con- ception of sets structured solely by equality (=), membership (2) and simple operations. By a critical review of Zermelo's system it begins the system- atic study of axiomatic set theory, which took the place of \naive" set theory. Fraenkel proposed the addition of the Axiom of Replacement to Zermelo's list. In 1923 von Neumann restored Cantor's theory of the transfinite by defining ordinal numbers using sets which are well-ordered by 2; he later introduced the Axiom of Foundation and promoted new hypotheses mediating towards Can- tor's Absolute. The axiomatic theory elaborated by Zermelo and Fraenkel (in- cluding Foundation) is called ZF and got great success as all interesting mathe- matical theories can be coded in ZF. Moreover, it is a theory with great interest in itself; infinity is no longer a metaphysical concept but just a definition han- dled through (finite) strings of characters in a prescribed language. A lack of Axiomatic set theory & unincreasable infinity 6727 ZF resides in the fact that it doesn't claim that two given sets are always com- parable (by Cantor's way); this claim is equivalent to the statement that for ev- ery set x there is a choice function on x (i.e., a function f such that f (y) 2 y for all non-empty y 2 x), so, it is achieved by adding to ZF the Axiom of Choice, and the resulting theory is called ZFC. This is by far the most important ax- iomatization of set theory. However, in ZFC not all sets we wish do exist. In particular, because of the above mentioned paradox, we are not allowed to consider all sets together. Similarly, a group theorist living in ZFC world is not allowed to handle the collection of all abelian groups (this is not a serious prob- lem, but somehow annoying!). A further lack of ZFC is that it is not finitely axiomatizable1, that is, its list of axioms cannot be written on a real sheet of pa- per. But we shall see that this is not a major problem as one can \embed" ZFC in a good way in a finitely axiomatizable theory called NBG after the names of von Neumann, Bernays and G¨odel.Moreover, switching to NBG will allow us to recover Cantor's concept of the Absolute in a fully consistent manner. 2 Set theory with classes The @α hierarchy of infinite cardinals is defined by transfinite recursion: •@ = ! is the smallest infinite cardinal. 0 + •@α+1 = (@α) , the successor cardinal to @α. •@λ = supα<λ @α for limit ordinals λ. Indeed, in ZFC every infinite set has some @α as cardinal number. Thus, the increasing sequence h@α : α 2 ONi, where ON denotes the collection of ordi- nal numbers, enumerates all infinite cardinalities. Looking at the size of the totality of these infinities, we can immediately realize that it is itself infinite: th considered the α infinite cardinal @α, the Power Set Axiom ensures the ex- @α istence of another cardinal number, namely 2 = @β, which is strictly larger than @α (by Cantor's theorem on the cardinality of the power set of a set). It follows that there is no greatest cardinal number, thus, there are infinitely many infinities. Moreover, they cannot all be collected together as elements of a single set, for no set can contain elements of all cardinalities: given any set S, each of its elements is a subset of S S = T, and so, 2jTj is a strict upper bound on the cardinalities of the elements of S. Therefore, the infinite collec- tion f@α : α 2 ONg is even \too big" to form a set, and hence larger than any of the infinities it enumerates (each of which is, by definition of cardinal number, the cardinality of a set): its magnitude remains outside the scope of the canonical axiomatic set theory ZFC. But can we go beyond the realm of the transfinite numbers @α, attempting to grasp this infinity? Although such a question transcends the ordinary set-theoretical framework provided by ZFC, in order to approach it we just need a shift of perspective, as emerging from the consideration of an alternative version of set theory due to 1As established by Montague in [5]. 6728 Raffaella Cutolo, Ulderico Dardano and Virginia Vaccaro von Neumann, Bernays and G¨odel,known as NBG. Unlike ZFC, whose domain is confined just to sets, two kinds of entities are distinguished in NBG: sets and proper classes (in general, we speak about classes). Proper classes are larger than sets and can never be elements of anything; so, while in ZFC we prove that the set of all sets doesn't exist, in NBG we can consider the proper class of all sets, the universe V = fx : x = xg. In fact, as we shall see, according to NBG's Class Comprehension Scheme all objects satisfying any given formula in the first-order language of NBG form a class: if not a set in ZFC, it is a proper class in NBG. Despite the differences regarding objects of study and axiomatization, NBG is a conservative extension of ZFC, meaning that: • the language of NBG extends the language of ZFC; • every theorem of ZFC is a theorem of NBG; • any theorem of NBG expressible using just the language of ZFC is al- ready a theorem of ZFC.2 Therefore, since a statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC, there is nothing to choose between ZFC and NBG as far as concerning sets.